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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimddvcbv | Structured version Visualization version GIF version |
Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 42570. The equivalent of this theorem without the bound variable change is rexlimddv 3155. Usage of this theorem is discouraged because it depends on ax-13 2371, see rexlimddvcbvw 42571 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rexlimddvcbv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) |
rexlimddvcbv.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) |
rexlimddvcbv.3 | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexlimddvcbv | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimddvcbv.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | |
2 | rexlimddvcbv.3 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | |
3 | rexlimddvcbv.2 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
4 | 2, 3 | rexlimdvaacbv 42570 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓)) |
5 | 1, 4 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 |
This theorem is referenced by: (None) |
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